# American Institute of Mathematical Sciences

August  2016, 21(6): 1917-1936. doi: 10.3934/dcdsb.2016029

## Nonlinear stability of stationary points in the problem of Robe

 1 Department Computer Science, University of Cincinnati, Cincinnati, Ohio 45221-0025, United States 2 Departamento de Matemática, Universidade Federal de Sergipe, São Cristovão-SE, CEP. 49100-000, Brazil

Received  August 2015 Revised  February 2016 Published  June 2016

In 1977 Robe considered a modification of the Restricted Three Body Problem, where one of the primaries is a shell filled with an incompressible liquid. The motion of the small body of negligible mass takes place inside this sphere and is therefore affected by the buoyancy force of the liquid. We investigate the existence and stability of the equilibrium points in the planar circular problem and discuss the range of the parameters for which the problem has a physical meaning.
Our main contribution is to establish the Lyapunov stability for the equilibrium point at the center of the shell. We achieve this by putting the Hamiltonian function of Robe's problem into its normal form and then use the theorems of Arnol'd, Markeev and Sokol'skii. Resonance cases and some exceptional cases require special treatment.
Citation: Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1917-1936. doi: 10.3934/dcdsb.2016029
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